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DARK MATTER, A NEW PROOF OF THE PREDICTIVE POWER OF GENERAL RELATIVITY

Stphane Le Corre (E-mail : le.corre.stephane@hotmail.fr)

No affiliation

Without observational or theoretical modifications, Newtonian and general relativity seem to be unable to explain gravitational behavior of large structure of the universe. The assumption of dark matter solves this problem without modifying theories. But it implies that most of the matter in the universe must be unobserved matter. Another solution is to modify gravitation laws. In this article, we study a third way that doesnt modify gravitation neither matters distribution, by using a new physical assumption on the clusters. Compare with Newtonian gravitation, general relativity (in its linearized approximation) leads to add a new component without changing the gravity field. As already known, this component for galaxies is too small to explain dark matter. But we will see that the galaxies clusters can generate a significant component and embed large structure of universe. We show that the magnitude of this embedding component is small enough to be in agreement with current experimental results, undetectable at our scale, but detectable at the scale of the galaxies and explain dark matter, in particular the rotation speed of galaxies, the rotation speed of dwarf satellite galaxies, the expected quantity of dark matter inside galaxies and the expected experimental values of parameters of dark matter measured in CMB. This solution implies testable consequences that differentiate it from other theories: decreasing dark matter with the distance to the clusters center, large quantity of dark matter for galaxies close to the clusters center, isolation of galaxies without dark matter, movement of dwarf satellite galaxies in planes close to the supergalactic plane, close orientations of spins vectors of two close clusters, orientation of nearly all the spins vector of galaxies of a same cluster in a same half-space, existence of very rare galaxies with two portions of their disk that rotate in opposite directions Keywords: gravitation, gravitic field, dark matter, galaxy.

1. Overview

1.1. Current solutions

Why is there a dark matter assumption? Starting with the

observed matter, the gravitation theories (Newtonian and general

relativity) seem to fail to explain several observations. The

discrepancies between the theory and the observations appear

only for large astrophysical structure and with orders of

magnitude that let no doubt that something is missing in our

knowledge of gravitation interaction. There are mainly three

situations that make necessary this assumption. At the scale of the

galaxies, the rotations speeds of the ends of the galaxies can be

explained if we suppose the presence of more matter than the

observed one. This invisible matter should represent more than

90% (RUBIN et al., 1980) of the total matter (sum of the observed

and invisible matters). At the scale of the clusters of galaxies, the

gravitational lensing observations can be explained with the

presence of much more invisible matter (ZWICKY, 1937; TAYLOR

et al., 1998; WU et al., 1998), at least 10 times the observed

matter. At the scale of the Universe, cosmological equations can

very well explain the dynamics of our Universe (for example the

inhomogeneities of the microwave background) with the presence

of more matter than the observed one. This invisible matter

should, in this third situation, represent about 5 times the

observed matter (PLANCK Collaboration, 2014).

How can we explain the origin of this dark matter? In fact, a more

objective question should be how we can explain these

discrepancies between theories and observations. There are two

ways to solve this problem, supposing that what we observe is

incomplete or supposing that what we idealize is incomplete. The

first one is to suppose the existence of an invisible matter (just like

we have done previously to quantify the discrepancies). It is the

famous dark matter assumption that is the most widely accepted

assumption. This assumption has the advantage of keeping

unchanged the gravitation theories (NEGI, 2004). The

inconvenient is that until now, no dark matter has been observed,

WIMPS (ANGLOHER et al., 2014; DAVIS, 2014), neutralinos (AMS

Collaboration, 2014), axions (HARRIS & CHADWICK, 2014). More

than this the dynamics with the dark matter assumption leads to

several discrepancies with observations, on the number of dwarf

galaxies (MATEO, 1998; MOORE et al., 1999; KLYPIN et al., 1999)

and on the distribution of dark matter (DE BLOK, 2009; HUI, 2001).

The second one is to modify the gravitation theories. The

advantage of this approach is to keep unchanged the quantity of

observed matter. The inconvenient is that it modifies our current

theories that are very well verified at our scale (planet, star, solar

system). One can gather these modified theories in two

categories, one concerning Newtonian idealization and the other

concerning general relativity. Briefly, in the Newtonian frame, one

found MOND (MILGROM, 1983) (MOdified Newtonian Dynamics)

that essentially modified the inertial law and another approach

that modifies gravitational law (DISNEY, 1984; WRIGHT & DISNEY,

1990). In the general relativity frame, one found some studies that

dont modify Einsteins equations but that looking for specific

metrics (LETELIER, 2006; COOPERSTOCK & TIEU, 2005; CARRICK &

COOPERSTOCK, 2010). And one found some other studies that

modified general relativity. Mainly one has a scalar-tensor-vector

gravity (MOFFAT, 2006) (MOG), a 5D general relativity taking into

account Hubble expansion (CARMELI, 1998; HARTNETT, 2006), a

quantum general relativity (RODRIGUES et al., 2011), two

generalizations of MOND, TeVeS (BEKENSTEIN, 2004; MCKEE,

2008) and BSTV (SANDERS, 2005), a phenomenological covariant

approach (EXIRIFARD, 2010)

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1.2. Solution studied in this paper

In our study, we will try a third way of explanation of these

discrepancies. This explanation doesnt modify the quantities of

observed matter and doesnt modify general relativity. To

demonstrate the capacity of the general relativity to explain dark

matter, we are going to use, in our paper, the native metric of

linearized general relativity (also called gravitoelectromagnetism)

in agreement with the expected domain of validity of our study

( and ). It doesnt modify the gravity field of

Newtonian approximation but defines a better approximation by

adding a component (called gravitic field, similar to magnetic field

of Maxwell idealization) correcting some imperfections of the

Newtonian idealization (in particular the infinite propagation

speed of gravitation). In a first paragraph, we will recall the

linearization of general relativity and give some orders of

magnitude of this component, the gravitic field. In a second

paragraph, we will focus first on the problem of the rotation of the

extremities of galaxies on sixteen galaxies. We will retrieve a

published result (LETELIER, 2006), showing that own gravitic field

of a galaxy cannot explain dark matter of galaxies. But one will see

that clusters gravitic field could replace dark matter assumption.

Before talking about its origin, one will see the orders of

magnitude of this component, expected to obtain the flat

rotation speed of galaxies. Its magnitude will be small enough to

be in agreement with the fact that it is undetectable and that the

galaxies are randomly oriented. In a third paragraph, we will talk

about the possible origins of this component. One will see that the

value of the gravitic field explaining dark matter for galaxies could

likely come from the cluster of galaxies (in agreement with a

recent observation showing that dark matter quantity decreases

with the distance to the center of galaxies cluster). In the next

paragraphs, one will use these values of gravitic field on several

situations. It will permit to retrieve many observations (some of

them not explained). These values (if interpreted in the hypothesis

of dark matter) will give the expected quantity of dark matter

inside the galaxies. These same values will explain the speed of

dwarf satellite galaxies. Its theoretical expression will explain their

planar movement (recently observed and unexplained). We will

also show that we can obtain a good order of magnitude for the

quantity of dark matter (obtained from the inhomogeneities

of the microwave background, CMB). One will still make some

predictions and in particular an original and necessary

consequence on the dwarf galaxies that can differentiate our

solution compared to the dark matter assumption. We will end

with a comparison between our solution and the dark matter

assumption. Some recent articles (CLOWE et al., 2006; HARVEY et

al., 2015) show that if dark matter is actually matter, it cannot be

ordinary matter. We will see at the end that these studies do not

invalidate the solution proposed in this paper. Instead they could

even allow testing our solution.

The goal of this study is to open a new way of investigation to

explain dark matter. This work has not the pretention to produce

a definitive proof (in particular it will need to be extended to the

general frame of general relativity and not only in its linearization

approximation). But it nevertheless provides a body of evidence

that confirm the relevance of the proposed solution and leads to

several significant predictions that differentiate it from the other

explanations.

To end this introduction, one can insist on the fact that this

solution is naturally compliant with general relativity because it is

founded on a component coming from general relativity (gravitic

field), traditionally neglected, on which several papers have been

published and some experimental tests have been realized. The

only assumption that will be made in this paper is that there are

some large astrophysical structures that can generate a significant

value of gravitic field. And we will see that the galaxies cluster can

generate it.

2. Gravitation in linearized general relativity

Our study will focus on the equations of general relativity in weak

field. These equations are obtained from the linearization of

general relativity (also called gravitoelectromagnetism). They are

very close to the modeling of electromagnetism. Let's recall the

equations of linearized general relativity.

2.1. Theory

From general relativity, one deduces the linearized general

relativity in the approximation of a quasi-flat Minkowski space

( ). With following Lorentz gauge, it

gives the following field equations (HOBSON et al., 2009)

(with

):

With:

The general solution of these equations is:

In the approximation of a source with low speed, one has:

And for a stationary solution, one has:

At this step, by proximity with electromagnetism, one traditionally

defines a scalar potential and a vector potential . There are in

the literature several definitions (MASHHOON, 2008) for the

vector potential . In our study, we are going to define:

With gravitational scalar potential and gravitational vector

potential :

With a new constant defined by:

This definition gives very small compare to .

The field equations can be then written (Poisson equations):

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With the following definitions of (gravity field) and (gravitic

field), those relations can be obtained from following equations:

With relations , one has:

The equations of geodesics in the linear approximation give:

It then leads to the movement equations:

One will need another relation for our next demonstration. In

agreement with previous Poisson equations , we deduce that

gravitic field evolves with ( ). More precisely, just like

in electromagnetism, one can deduce from Poisson equation

that

but it is its dependence in

that is

pertinent for our study.

From relation , one deduces the metric in a quasi flat space:

In a quasi-Minkowski space, one has:

We retrieve the known expression (HOBSON et al., 2009) with our

definition of :

Remark: Of course, one retrieves all these relations starting

with the parametrized post-Newtonian formalism. From

(CLIFFORD M. WILL, 2014) one has:

The gravitomagnetic field and its acceleration contribution

are:

And in the case of general relativity (that is our case):

It then gives:

And with our definition:

One then has:

With the following definition of gravitic field:

One then retrieves our previous relations:

A last remark: The interest of our notation is that the field

equations are strictly equivalent to Maxwell idealization. Only

the movement equations are different with the factor 4. But

of course, all the results of our study could be obtained in the

traditional notation of gravitomagnetism with the relation

.

To summarize Newtonian gravitation is a traditional

approximation of general relativity. But linearized general

relativity shows that there is a better approximation, equivalent to

Maxwell idealization in term of field equation, by adding a gravitic

field very small compare to gravity field at our scale. And, as we

are going to see it, this approximation can also be approximated

by Newtonian gravitation for many situations where gravitic field

can be neglected. In other words, linearized general relativity

explains how, in weak field or quasi flat space, general relativity

improves Newtonian gravitation by adding a component (that will

become significant at the scales of clusters of galaxies as we will

see it).

In this approximation (linearization), the non linear terms are

naturally neglected (gravitational mass is invariant and

gravitation doesnt act on itself). This approximation is valid

only for low speed of source and weak field (domain of validity

of our study).

All these relations come from general relativity and it is in this

theoretical frame that we will propose an explanation for dark

matter.

2.2. Orders of magnitude

The theory used in this study is naturally in agreement with

general relativity because it is the approximation of linearized

general relativity. But it is interesting to have orders of magnitude

for this new gravitic field.

2.2.1. Linearized general relativity and classical

mechanics

In the classical approximation ( ), the linearized general

relativity gives the following movement equations ( the inertial

mass and the gravitational mass):

A simple calculation can give an order of magnitude to this new

component of the force due to the gravitic field. On one hand,

gravity field gives

, on the other

hand, for a speed (speed of the source that

generates the field) one has

. That is to say that for a test particle

speed , the gravitic force compared to the

gravity force is about . This new term is

extremely small and undetectable with the current precision on

Earth.

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2.2.2. Linearized general relativity and special

relativity

Linearized general relativity is valid in the approximation of low

speed for source only. But, there isnt this limitation for test

particle. One can then consider a test particle of high speed. In the

special relativity approximation ( ), with the speed of the

test particle, the linearized general relativity gives the following

movement equations:

It is the same equation seen previously, with here the relativistic

momentum. We have seen that in classical

approach. If (speed of the test particle), one always

has which is always very weak compared to

the force of gravity ( ). Once again, even in the

domain of special relativity (high speed of test particle), the

gravitic term is undetectable.

To explain dark matter, we are going to see that we need to

have inside the galaxies. It means that, for our galaxy,

our solar system must be embedded in such a gravitic field. For a

particle test of speed , it gives a gravitic

acceleration . Compared to gravity

acceleration ( ) near Sun, it represents an

undetectable correction of about (on the deviation of

light for example).

3. Gravitic field: an explanation of dark matter

One of clues which push to postulate the existence of dark matter

is the speed of the ends of the galaxies, higher than what it should

be. We will see that the gravitic field can explain these speeds

without the dark matter assumption. For that, we will first

consider an example (data from Observatoire de Paris) to

demonstrate all our principles. Next, we will apply the traditional

computation (KENT, 1987) on sixteen measured curves: NGC 3198,

NGC 4736, NGC 300, NGC 2403, NGC 2903, NGC 3031, NGC 5033,

NGC 2841, NGC 4258, NGC 4236, NGC 5055, NGC 247, NGC 2259,

NGC 7331, NGC 3109 and NGC 224.

3.1. Dark matter mystery

Some examples of curves of rotation speeds for some galaxies are given in Fig. 1.

One can roughly distinguish three zones:

Zone (I) [0 ; 5 kpc]: close to the center of the galaxy, fast growth rotation speed

Zone (II) [5 ; 10 kpc]: zone of transition which folds the curve, putting an end to the speed growth

Zone (III) [10 kpc ; ..[: towards the outside of the galaxy, a flat curve with a relative constancy speed (contrary to the decreasing theoretical curve)

Gravitation, for which speed should decrease, failed to explain such curves in zone (III) without dark matter assumption. And these examples are not an exception; it is a general behavior.

3.2. Basis of our computation

The dark matter assumption is a way to increase the effect of the

gravity force. To be able to have a gravitic force ( )

that can replace the dark matter, the gravitic field should have a consistent orientation with the speed of the galactic matter to generate a centripetal force (just like the gravity force). This can be performed for example with the situation of the following figure:

The simplified Fig. 2 can help us to visualize how the two components of the linearized general relativity intervene in the equilibrium of forces. The gravitic field, perpendicular to galaxy rotation plane, with the velocity of the matter generates a centripetal force increasing the Newtonian gravitation.

A first non trivial point should be explained: the orientation of

generates a centripetal force (and not a centrifugal one). At this

step, one can consider it as an assumption of our calculation. But

we will see that with our solution, cant come from galaxies.

Then, another origin will be studied. And we will see that, in this

context, the centripetal orientation can be justified (5.3), and

furthermore it will lead to very important sine qua none

predictions that will differentiate our solution from the traditional

dark matter assumption and from MOND theories. So this

assumption will be only a temporary hypothesis. It will be a very

constraining non trivial condition, but some observations seem to

validate it.

A second non trivial point to explain is that the gravitic field is perpendicular to the speed of the matter. It can be justified independently of its origin. Our solution, for the galaxies, represents an equilibrium state of the physical equations. This means that the planes of movement must be roughly

perpendicular to . It can be explained by a filtering of the speed over the time. If, initially, the speed of a body is not perpendicular

to the gravitic field , the trajectory of this body wont be a closed trajectory (as a circle or an ellipse) but a helicoidally trajectory, meaning that this body will escape. Finally, at the equilibrium state (state of our studied galaxies), the gravitic field will have

Galaxy (schematically)

Fig. 2: Simplified representation of the equilibrium of forces in a galaxy

Fig. 1: Superposition of several rotation curves (Sofue et al., 1999)

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imposed the movement of the matter in roughly perpendicular

planes to . One can add that can change its direction with without modifying this expectation. This allows posing that in our calculations, we are in the very general

configuration whatever the position .

The traditional computation of rotation speeds of galaxies consists in obtaining the force equilibrium from the three following components: the disk, the bugle and the halo of dark matter. More precisely, one has (KENT, 1986):

Then total speed squared can be written as the sum of squares of

each of the three speed components:

From this traditional decomposition, we obtain the traditional

graph of the different contributions to the rotation curve (just like

in Fig. 3).

Disk and bulge components are obtained from gravity field. They

are not modified in our solution. So our goal is now to obtain only

the traditional dark matter halo component from the linearized

general relativity. According to this idealization, the force due to

the gravitic field takes the following form

and it corresponds to previous term

. I recall

that we just have seen that the more general situation at the

equilibrium state is to have the approximation . In a second

step, we will see some very important consequences due to the

vector aspect ( ), it will lead to several predictions. This

idealization gives the following equation:

Our idealization means that:

The equation of dark matter (gravitic field) is then:

This equation gives us the curve of rotation speeds of the galaxies

as we wanted. The problem is that we dont know this gravitic

field , but we know the curves of speeds that one wishes to have.

We will thus reverse the problem and will look at if it is possible to

obtain a gravitic field which gives the desired speeds curves.

From the preceding relation , let us write according to ,

one has:

3.3. Computation step I: a mathematical solution

To carry out our computation, we are going to take in account the

following measured and theoretical curves due to the

Observatoire de Paris / U.F.E. (Fig. 3).

On Fig. 4, one has the gravitic field computed with formula . The numerical approximation used for and curves are given at the end of the paper in Tab.2:

This computed gravitic field is the one necessary to obtain the

measured rotation speed of this galaxy without dark matter

assumption. This curve plays the same role than the distribution

of dark matter in the eponym assumption, it explains . But of

course one must now study this solution in a more physical way.

We will see that if we only consider the internal gravitic field of

the galaxy it doesnt physically work but a solution can be

imagined. This solution will be able to justify physically this curve,

to be in agreement with observations and even to explain some

unexplained observations.

3.4. Computation step II: a physical solution

At this step, we only mathematically solved the problem of the

dark matter by establishing the form that the gravitic field

should take. The only physical assumption that we have made is

that the force due to the gravitic field is written in the form

. And under this only constraint, we just come to show that it

is possible to obtain a speeds curve like that obtained in

experiments. To validate this solution, it is necessary to check the

physical relevance of this profile of gravitic field and in particular

to connect it to our gravitic definition of the field. Furthermore, at

this step, with our approximation, this computed gravitic field

doesnt represent only the gravitic field of the galaxy but also the

Fig. 4: Gravitic field which gives the expected rotational speed curve of Fig. 3.

Fig. 3: Rotation curve of a typical spiral galaxy, showing measurement points with their error bar, the curve (green) adjusting the best data (black), the speed of the disk (in blue) and that of a halo of invisible matter needed to account for the observed points (in red). [Crdit "Astrophysique sur Mesure"]

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others internal effects, in particular near the center (as frictions

for example). But our goal will be to obtain a good physical

approximation far from the galaxy center where the dark matter

becomes unavoidable.

To physically study this solution far from galaxy center, the galaxy

can be approximated as an object with all its mass at the origin

point and some neglected masses around this punctual center.

Such an idealization is an approximation in agreement with the

profile of the mass distribution of a galaxy. Far from galaxy center,

its also the domain of validity of linearized general relativity. Far

from galaxy center, one then should retrieve some characteristics

of our punctual definition (

in

agreement with Poisson equations) in particular the three

following characteristics: decreasing curve, curve tending to zero

and

.

Positive point: To be acceptable physically, it is necessary at least

that this field is decreasing far from its source. For a galaxy, the

large majority of its mass is in the neighborhoods of the center

(zone (I)). Thus one must obtain a gravitic field which decreases

far from the central area (zone (III)). The field obtained is globally

in conformity with this expectation.

Negative points: Compared to the characteristic of the punctual

definition (

), the previous gravitic field

reveals two problems. First, our curve doesnt decrease near the

galaxys center. Secondly, our curve doesnt decrease to zero.

Lets see the first problem: The gravitic field graph does not

always decrease. The curve starts to decrease only around .

This problem can certainly be solved because our approximation

take in account only gravitation effect. More precisely, gravitic

field curve is computed starting with real value of rotation speed.

But this measured rotation speed is due to gravitation but also to

others phenomena (frictions for example). And our computation

procedure takes in account only gravitation. So, previous

computed curve (obtained from measured speeds) doesnt

represent only gravitation, in particular in zone (I) and (II). In these

zones, the dynamic is dominated by others phenomena due to

density of matter, in particular frictions and collisions, that could

explain the graph. More than this, in these zones, the condition of

validity of the approximation of general relativity in a quasi flat

space is certainly not verified. So in this zone, our approximation

underestimates the real internal gravitic field (computed value =

real value - dissipation due to the frictions effects).

Lets see now the second problem which will lead us to a possible

explanation of the dark matter. The end of previous computed

gravitic field curve should tend to our punctual definition of

gravitic field. That is to say that gravitic field should decrease to

zero. It clearly does not. More precisely, it should decrease to zero

with a curve in

. One can try to approximate previous computed

curve with a curve in

(Fig. 5). But unfortunately, the

approximation is so bad that it cannot be an idealization of the

real situation as one can see it on the following graph.

This result means that the internal gravitic component

(internal because due to the own galaxy) cannot explain the disk

rotation speed at the end of the galaxies. One can note that this

result is in agreement with the studies in (LETELIER, 2006),

(CLIFFORD M. WILL, 2014) or (BRUNI et al., 2013) that also take

into account non linear terms. It also means that, far from the

center of the galaxy, even the non linear term cannot explain the

dark matter.

But this gravitic field is generated by all moving masses. The

clusters of galaxies, the clusters of clusters (superclusters) and so

on have a gravitic field. The previous calculation only invalidates

the galactic gravitic field as an explanation of dark matter. And as

a galaxy is embedded in such astrophysical structures (cluster,

supercluster), one has to study if the gravitic field of these large

structures could explain the rotation speeds of the ends of the

galaxies. Instead of studying each structure one by one (that

would be difficult because we dont know the value of their own

gravitic fields) one simplifies the study by looking at, in a more

general way, what value of gravitic field one should have to

explain the rotation speed of the ends of galaxies. By this way, one

will have the order of magnitude of the required gravitic field.

Then, one will be able to show which large astrophysical

structures could give such a value or not. We will see that the

clusters (and their neighbors) are very good candidates. And also,

from these values of gravitic field, one will be able to obtain lot of

very interesting observational results and to make predictions to

test our solution.

So, because general relativity implies that all large structures

generate a gravitic field and because observations imply that

galaxies are embedded in these large structures, we make the

unique assumption of our study (and once again we will see that

this assumption will be more likely a necessary condition of

general relativity):

Assumption (I):

Galaxies are embedded in a non negligible external gravitic

field

This external gravitic field, as a first approximation, is locally

constant (at the scale of a galaxy).

Remarks: The assumption is only on the fact that the gravitic field

is large enough (non negligible) to explain rotation curve and not

on the existence of these gravitic fields that are imposed by

general relativity (Lense-Thirring effect). And one can also note (to

understand our challenge) that the constraints imposed by the

observations imply that, in the same time, it must be small enough

not to be directly detectable in our solar system and not to

impose the orientation of the galaxies but large enough to explain

Fig. 5: Approximation of gravitic field with a curve in . It cannot represent reality.

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dark matter. All these requirements will be verified. Furthermore,

when we are going to look at the origin of this embedding gravitic

field, this assumption will become more an unavoidable condition

imposed by general relativity at the scale of clusters than a

hypothesis. The second point is mainly to simplify the

computation and we will see that it works very well.

But of course, this external gravitic field should be consistently

derived from the rotation of matter within general relativity (it will

be seen in 4.1). But we will also see (6.2) that this

approximation of a uniform gravitic field is compliant with

linearized general relativity (just like it is in Maxwell idealization

for electromagnetism).

Mathematically, this assumption means that

with the internal gravitic component of the

galaxy

(punctual definition of the linearized

general relativity) and an external gravitic component , the

approximately constant gravitic field of the close environment

(assumption (I)). If we take and

(values obtained to adjust this curve on the previous computed

one), one has the curves on Fig. 6:

Left curves represent gravitic field and right curves the rotation

speed obtained from formula . In the last part of the graph

(domain of validity of linearized general relativity), the curves are

indistinguishable. From about to the end, the evolution in

is excellent.

Important remark: To be clearer, our expression of dark matter

is . To idealize our curve

with as previously, it doesnt imply that the

internal gravitic field of the galaxies

and the external gravitic

field are collinear. Indeed, as one can see on the values of

these two components, except in a transition zone (around a

value noted , as we are going to see it below, leading to very

important predictions), the two vectors can be seen as spatially

separated,

for and for .

Concretely, it gives for

because in this area at the equilibrium. I recall

that, at the equilibrium (over the time), has imposed the

movement of the matter in roughly perpendicular plane to .

And for because in

this area at the equilibrium. So the vectors dont need

to be collinear to write . We will

be in this situation for all the next studied galaxies (with two

components very different in magnitude that allows seeing them

as spatially separated). Our approximation is then very general,

except in a transition zone that is not studied in our paper.

One can also add that the order of magnitude of and

far

from the center of the galaxy allows justifying the use of linearized

general relativity.

To summarize our dark matter explanation, in the frame of

linearized general relativity, the speeds of rotation at the ends of

the galaxy can be obtained with:

An internal gravitic field that evolves like far from the

center of galaxy (in agreement with the punctual definition

of linearized general relativity)

A constant external gravitic field embedding the galaxy.

One can write these contributions:

Remark: In theoretical terms, the use of the approximation

allows extracting the pure internal gravitic field by neglecting the

other non gravitic effects from the central area of the galaxy. But

as it is obtained by fitting experimental curves that underestimate

gravitational effects (because of the frictions for example, as said

before), this term must always be underestimated. We will

quantify this discrepancy. I recall that our main goal is to obtain an

approximation in the ends of the galaxy (that is to say ), area

where the other non gravitic effects vanish. Our approximation

should be sufficient for this goal.

One can note that this contribution to be compliant with general

relativity implies two constraints. First, the term

that is the own

(internal) gravitic field of the galaxy should be retrieved from

simulations ever done in several papers (for example BRUNI et al.,

2013). It will validate our approximation. Secondly, for each

gravitic field, there should be a gravity field. For example, in our

computation, associated to

, we take into account the gravity

field for the galaxy (the term

). But for our external

gravitic field , there should also be a gravity field. So to be

compliant with our assumption, this gravity field should be

negligible compare to (because we dont take it into account).

These constraints will be verified in this study.

Remark 3: At the end of our study, we will generalize the

expression for a source of gravitic field that wont be

punctual but a gravitic dipole (just like in electromagnetism,

with a magnetic dipole).

3.5. Application on several galaxies

We are now going to test our solution on different galaxies

studied in (KENT, 1987). Because the linearized general relativity

doesnt modify the components and , one can

focus our study on the relation .

For that we are going to determine curve, from

experimental data with the relation, formula :

Fig. 6: Gravitic field approximation (

) and the

rotation speed computed with it.

8

I recall that we know the curve which is given by

experimental data and we know , the resultant

component obtained from relation

where and

are also

deduced from experimental data. The numerical approximation

used for and curves are given at the end of the

paper in Tab. 3.

Then we are going to approach this curve (blue curves in

Fig. 7) with our expected expression which

explains dark matter (black curves of right graphs in Fig. 7).

9

10

11

Discussion on these results:

1) Our solution makes the assumption of a constant asymptotic

behavior of gravitic field (tending to the external gravitic field

at the ends of galaxies). These results confirm the tendency to get

a curve flattens for large . However, there are two

exceptions. The last two curves do not flatten in the studied

interval of distance. These results also show that the external

gravitic field values are in the interval .

2) Our solution makes the more precise assumption of an

evolution of the field as far from the center of the galaxy.

This hypothesis is verified because theoretical and

experimental curves are equivalent on more than 30 % of

their definition (up to 50 % for some), always with the exception

of the last two curves. Furthermore, our idealization comes from

linearized general relativity approximation that is valid only in

zone (III) far from the center of the galaxy and then nearly 100% of

the zone (III) is explained. This is the first main result of our study.

These results also show that, with our approximation, the internal

gravitic field values are in the interval . We

are going to compare these values with published studies.

3) One can note this remarkable agreement with observations.

The order of magnitude of is small enough to be undetectable

in our solar system (cf. the calculation in paragraph 2) and at the

same time it is high enough to explain rotation speed of galaxies.

Dark matter has the disadvantage of being almost all the matter of

our universe and to be undetectable until now, meaning that all

our theories would explain (and would be founded on) only a

negligible part of our reality.

4) Validity of our approximation: The value is associated to the

internal gravitic field of galaxies. As we said before, it has ever

been computed in several studies. One can then compare our

approximation to these more accurate studies. For example, NGC

3198 gives the value for a visible mass of

about (VAN ALBADA et al., 1985). One then has:

As seen at the beginning of our study, in term of

gravitomagnetism, one has , it leads to a ratio

. In (BRUNI et al., 2013), the simulation gives a

ratio between the vector and scalar potential of about . As

we said previously, our deduced internal gravitic field must be

underestimated compare to the real value because our

idealization doesnt take into account the frictions effects

(computed value = real value - dissipation due to the frictions

effects). And secondarily, our idealization doesnt take into

account the non linear terms. It then explains that our ratio is

slightly inferior to a more accurate simulation. But even with that,

our approximation gives a good order of magnitude of the relative

values of the gravity and gravitic fields. If one looks at the absolute

value of our gravitic field, at , one has

. In (BRUNI et al., 2013), the simulation

gives for the power spectra the value . This

comparison confirms that our approximation gives a gravitic field

that is underestimated. With our example, the factor of correction

(due mainly to frictions effects) is about . Its effective

gravitic field is around 5 times greater than our approximation.

But the order of magnitude is still correct (even without the non

linear terms). Lets make a remark. Our explanation of dark matter

is focused on the external term . The next results will be

Fig. 7: Left graphs are the basis data from (KENT, 1987); right graphs represent (in blue) and

(in black)

explaining ends of rotation speed curves (analytic expression of is given).

12

obtained far from the galaxies center. We are going to see that

far from the center of galaxies only acts. And far from the

center of galaxies, the non linear terms (of internal gravitic field)

and frictions effects are negligible. For these reasons, with our

approximation, the value of external gravitic field should be a

more accurate value than our .

5) A first reaction could be that the external gravitic field should

orient the galaxies in a same direction, which is at odds with

observations. With our analysis, one can see that the internal

gravitic field of galaxies (

) is bigger than the embedding

gravitic field ( ) required to explain dark matter

for (

). And if one

takes in account the factor of correction seen previously on ,

the internal gravitic field imposes its orientation until .

With these orders of magnitude, we understand that galaxies

orientation is not influenced by this gravitic field of the

environment. The randomly distribution of the galaxies

orientation is then completely in agreement with our solution. We

will see further that our solution implies in fact a constraint on the

distribution of the galaxies orientation. Inside a cluster, galaxies

orientation will have to be randomly distributed in a half-space. It

will be an important test for our solution.

6) Influence of internal gravitic field far from the galaxys center:

One can note that, for only the internal gravitation, one has in the

galaxies

. And for , one has in the

galaxies . It implies that

in this area. So, far

from galaxies center, the gravitic force is of the same magnitude

than the gravity force. And then, just like the internal gravity

force, the internal gravitic force decreases and vanishes far from

galaxies center explaining why it cannot explain dark matter.

At this step, to compare in the main lines and in a simplified way

our study with already published papers, one can say that in terms

of linearized general relativity the already published papers

idealize the gravitic field like

(but more accurately, taking

into account non linear terms) and in our approach we idealize the

gravitic field like . The already published papers

deduce their expression from mass distribution of galaxies, giving

them only the gravitic field of galaxies. Conversely, we deduce our

expression from the rotation speeds of galaxies, giving us the

gravitic field of galaxies and a supplemented external gravitic field.

The rest of our study is going to justify this expression by showing

that the computed values (for ) can be deduced from larger

structures than galaxies; that the computed values (for ) can

explain unexplained observations and retrieve some known

results; by showing that our expression is consistent with general

relativity. It means to show that the uniform verifies the field

equations of linearized general relativity (that is our domain of

validity) and that has to be associated with a negligible gravity

field .

4. Cluster as possible origin of the gravitic field

We are now going to use these values of (

). It will allow obtaining the expected rotation speed of

satellite dwarf galaxies and retrieving the expected quantities of

dark matter in the galaxies. We will also demonstrate that the

theoretical expression of gravitic field can explain the dynamic of

satellite dwarf galaxies (movement in a plane) and allow obtaining

an order of magnitude of the expected quantitiy of dark matter

in the CMB. But just before we are going to show that the clusters

of galaxies could generate these values of .

4.1. Theoretical relevancies

Before beginning this paragraph, I would like to precise that the

goal of this paragraph is not to obtain a value of the gravitic field

for large structures but to determine which large structure(s)

could be a good candidate(s) to generate our embedding gravitic

field. Inside our theoretical frame, we determined the values of

the internal gravitic fields of the galaxies. From these values and

from the definition of the gravitic field (which depends on the

mass and the speed of source), we are going to apply a change of

scale to estimate the value of the internal gravitic field of objects

larger than galaxies. Of course, this way of approximating can be

criticized. But because the same criticism can be assigned to each

large structure, one can then hope, by this way, obtaining the

relative contribution of these large structures (but not necessarily

their actual values). This process will show that the galaxies'

clusters are certainly the main contributors to our embedding

gravitic field (and astonishingly this way of calculation will also

give the good order of magnitude of the expected gravitic field,

i.e. not only in relative terms).

About the mean value :

What can be the origin of this embedding gravitic field? In our

theoretical frame, the gravitic field (just like magnetic field of a

charge in electromagnetism) can come from any moving mass.

Because of the orders of magnitude seen previously (for particle,

Earth, Solar system and galaxies) one can expect that the

embedding gravitic field ( ) should be due to a very large

astrophysical structure (with a sufficient internal gravitic field

to irradiate large spatial zone). Furthermore, just like magnetic

field of magnets (obtained as the sum of spins), this external

gravitic field could also come from the sum of several adjacent

large structures. Our theoretical solution then implies that our

embedding gravitic field can come from the following possibilities:

Case A:

Internal gravitic field of the galaxy

Case A bis:

Sum of internal gravitic fields of several close galaxies

Case B:

Internal gravitic field of the cluster of galaxies

Case B bis:

Sum of internal gravitic fields of several close clusters

Case C:

Internal gravitic field of larger structure (cluster of cluster,)

Case C bis:

Sum of internal gravitic fields of several close larger structures

CaseD:

Internal gravitic field of our Universe

13

This expected embedding gravitic field should be consistently

derived from the rotation of matter within general relativity. It

means that if it should come from an internal gravitic field of a

large structure, it should evolve like

far from the source (in

agreement with linearized general relativity because of Poisson

equations ). So, lets see how large can this term

be for

these large structures.

Case A: We have seen that our solution (in agreement with some

other published papers) rejects the possibility of an internal

gravitic field of galaxy sufficient to explain dark matter. Lets

retrieve this result. We are going to see that, in our

approximation, around , the internal and external

gravitic fields have similar magnitudes. But for , the

internal gravitic field of the galaxy cant explain because it

becomes too small. And we are going to see that the value

of can explain the rotation speed of satellite dwarf

galaxies (for ). But with an internal gravitic field of

galaxy about , at , one then has

the value . At internal gravitic field of galaxy

represents only about 1% of (for an average value

of ). Effectively, the gravitic field of a galaxy cant

explain .

Case A bis: Even the sum of several close galaxies cant generate

our required . Roughly in a cluster of galaxies, there are

about galaxies for a diameter of . It

leads to an average distance between galaxies of

about (for an idealization as a disk:

).

The contribution to the gravitic field of an adjacent galaxy at the

distance

with

is then

. If we suppose that there are six neighboring galaxies (two

galaxies by spatial direction) it leads to a gravitic field of

about . It only represents % of the expected external

gravitic field.

Case B: Without an explicit observation of the internal gravitic

field of a cluster, it is difficult to give its contribution to the

embedding gravitic field . One can try a roughly approximation.

The mass of a cluster is about (without dark matter). It is

about 1000 times the mass of a typical galaxy ( ) and the

typical speed of the matter inside the cluster compare to its

center are around 5 times greater than matter in galaxies (in our

calculation, it is not the peculiar speed). By definition, the gravitic

field is proportional to the mass and to the speed of the source

(

). If we assume that the factor of

proportion for matter speed is also applicable for the speed of the

source, the internal gravitic field of a cluster could then be

about

. With a typical clusters

size of around , it gives at its borders the

following value of gravitic field

. It then

represents around 20% of the expected external gravitic field .

As in the precedent case A bis, it is not sufficient to obtain , but

contrary to the case A bis, more we are close to the clusters

center, more the external gravitic field could be entirely explained

by internal gravitic field of the cluster (for example at 700kpc it

represents about 100%).

Case B bis: Furthermore, the sum of the internal gravitic field of

adjacent clusters could maintain this embedding gravitic field on

very large distance. For example in the case of the Local Super

Cluster, there are around clusters for a diameter

of . It leads to an average distance between

clusters of about (for an idealization as a disk:

). We have seen that roughly internal gravitic field of a

cluster would be about . The contribution to the

gravitic field of an adjacent cluster at the distance

is then

. If we suppose that a

cluster has around six close neighbors. It leads to a gravitic field of

about . It represents 100% of the required external gravitic

field.

Case C: The mass of a supercluster as the Local Super Cluster is

about without dark matter ( with dark matter). It

is around times the mass of a typical galaxy ( ). Lets

take a typical speed of the matter inside the supercluster compare

to its center around 5 times greater than matter in galaxies (for

Laniakea, the velocities of the matter compare to its center are

only around ). By definition, the gravitic field is

proportional to the mass and to the speed. It gives then roughly

that the internal gravitic field of a supercluster could be

about . With (size

of a supercluster) it would give

. It

then represents around 2% of the expected external gravitic field.

Case C bis: Even if one considers ten neighbors for this

supercluster, one cannot obtain the expected value of .

A first conclusion is that with our very elementary analysis (by

keeping the proportionalities between different scales), the

cluster with its first neighbors could be a relevant structure to

generate our embedding gravitic field . Inside the cluster, the

order of magnitude is correct and more we are close to its border

more the contribution of its first neighbors compensate it. But one

cannot exclude the possibility that a supercluster could generate a

more important gravitic field than what we have computed. In fact

there are others reasons that could make the superclusters less

appropriate to explain our . Lets see these reasons. We just

have seen that between several astrophysical structures (larger

than galaxies), the clusters seem to be the structures that

generate the more important mean value of (and

surprisingly, it also gives the good order of magnitude). We are

now going to see that the interval of expected (not only the

mean value as before) also leads to the same conclusion.

About the interval :

As we are going to see below, very far from the galaxys center,

our explanation of the dark matter is given only by the

relation (with the distance to the galaxys center). In

this area, one then has a relation between the speed and the

embedding gravitic field (for a fixed position , distance to the

galaxys center). Hence, this relation can allow analyzing our

solution in distances terms. Indeed, one can look for the extreme

14

positions (distance inside a structure, relatively to its center)

that allow obtaining the extreme values .

Here is the distance from the structures center and not the

distance that appears in the equilibriums equation used in our

study. If we take, for clusters example, for its internal

gravitic field, one obtains at ,

and at , . So,

the studied galaxies should be situated in an interval of size

around for a typical clusters size of around 2.5Mpc to

obtain our values of (

). Because nearly

all the known galaxies have dark matter and that nearly all the

known galaxies are in clusters, a representative sample of galaxies

should lead to an interval of, at least, the size of the cluster. From

this approach, one can then refine our solution. Because our

studied galaxies are in several clusters, and because of the

diversity of these clusters, the size of this previous interval

(around ) is likely too small than expected (size of the

cluster). If we look at the rotations speeds curves of our studied

galaxies, the main interval is around

. If we look at the rotations speeds curves of all

known galaxies, the main interval is at least

. One can then expect that our studied galaxies

underestimate the interval of around a factor 2. But even with this

correction, the interval increases only of a factor (because

of an evolution in ) given then an interval of around . It

is always too small to be representative (20% of the expected

interval).

If the source was the gravitic field of the supercluster, the interval

needs to have a typical size of around the size of the supercluster

(ten times greater than for the cluster, 25Mpc). Our previous

calculated interval would then represent only 2% of the expected

interval. The situation is more critical for the supercluster.

How can we enlarge our interval? A way could be to increase the

value of . Unfortunately, this interval doesnt notably increase

with the value of whatever the considered structure. At this

step, one can once again note that the cluster is still clearly the

candidate that gives the greater interval, but it gives a too small

interval (20%). Another way could be to consider an evolution

in with for the gravitic field. But first, in relative terms,

it applies the same correction to all the structures (it doesnt then

modify the best candidate). And secondly, even with , the

interval, for the cluster, still only represents 25% of the expected

interval.

Consequently, in our solution, the more likely way to maintain

on larger distance is only a case of kind Bis (our previous case B

bis, C bis). In other words, our solution implies that the gravitic

field of a large structure must be compensated by its neighbors at

its border. With our previous calculations, the gravitic field of a

cluster could be compensated by the gravitic field of its nearest

neighbors, but for the supercluster, it seems unlikely. It would

need around 50 superclusters neighbors (which is not possible) to

compensate the littleness of the value of at its border. For the

supercluster, the only way to decrease the number of neighbors

would be to have a greater (not to enlarge the interval but to

have a greater at its border). But then the expected interval

of would be moved to its border without being notably

enlarged. It would then imply the existence of lots of galaxies with

very much more dark matter for a large area close to the

superclusters center (consequence of an increase of ). The

observations dont show such a situation. It is then unlikely that a

supercluster has a so great internal gravitic field.

By this way, once again only the clusters (with its neighbors) seem

to be able to generate the expected on large areas. Indeed,

from 900kpc to the clusters center, we start to be in the expected

interval of . And when we are close to the clusters border, the

neighbors compensate the internal (until 2.5Mpc as seen

before). It then gives a size of 1.6Mpc. But furthermore,

symmetrically, the expected value of still goes on in the

neighbors. In other words, between two clusters centers (around

5Mpc), one has an interval of expected along around 3.2Mpc. It

then represents 65% of the expected typical size of interval. And if

we apply the correction of (for a more generic sample), it

leads to 90% of the expected interval. The 10% left should

correspond to an area near the clusters center where there

should be more dark matter (in agreement with the observations

as we are going to see it).

A second conclusion is that in our solution, the superclusters are

irrelevant to generate a sufficient spatial extension for our . And

once again the clusters with their nearest neighbors can generate

it (with this time an astonishing good order of magnitude for the

spatial extension of ).

About the statistic of the dark matter inside the galaxies:

In a complementary way, one can deduce that if the considered

structure was larger than a supercluster of kind of Local Super

Cluster (as Laniakea for example) and if a mechanism to

maintain would be possible on large area (which seems hard to

realize as seen previously), the existence of galaxies without dark

matter would be very unlikely. The observations show that there

are few galaxies without dark matter. Once again, it seem unlikely

that larger structures than superclusters can be consistent with

our solution.

Inversely, if was explained by only one cluster (without its

neighbors), one should find lot of galaxies without dark matter. In

fact, all galaxies at the border of a cluster would be without dark

matter. The observation doesnt show such a situation.

A third conclusion is that in our solution, larger structures than

superclusters and only cluster (without its neighbors) are

irrelevant to generate our and to be in agreement with

observation. 0nce again clusters with their nearest neighbors can

generate it and be in agreement with the observations.

About the negligible :

Furthermore, the cases B and B bis are important and relevant

because they allow justifying our second constraint on assumption

(I). We have seen that there are two constraints on our

assumption to be compliant with general relativity. The first one

(to retrieve the already calculated and published term

) was

verified previously. The second one is on the fact that there

should be a weak gravity field associated with the gravitic

field . We have seen that the influence of the gravitic field

evolves like

. In the galaxies, for one

has . It implies that, in this area (ends of

15

galaxies)

. For the internal gravitational fields of

galaxies, one has seen that one has around

(giving

as ever said). But in the case B of a single cluster,

the typical speed is about 5 times greater than in the galaxies. So,

for the internal gravitational fields of clusters, one could

have

(because the gravitic field

depends on the sources speed). It then gives in the context of the

ends of galaxies

. It means that the component of gravitic

field of the cluster represents more than 80% of the gravitational

effect due to the cluster. In an approximation of first order, the

gravity field of the cluster can be neglected (

). Furthermore, one can add that this order of magnitude is

in agreement with the observations.

With a Newtonian approximation, one can show the consistency

of this origin. The gravitation forces (due to the cluster)

are

. This can be interpreted as a modification of the

observed mass of the cluster

. One

has seen that for the clusters

at the ends of the galaxies

(because of the dependency with the speed). One then retrieves

roughly the expected quantity of dark matter, a little less than ten

times greater than what it is observed.

With the case B bis, there is a second reason to neglect the gravity

field . Associated with the case B bis, our mechanism that

could explain the spatial extension of could also allow

neutralizing the gravity field. If we take into account the clusters

neighbors, the gravity field must greatly decrease and the gravitic

field can increase in the same time, as one can see on this

simplified representation (Fig. 8).

The sum of these vectors allows justifying (once again) that the

associated gravity field could be neglected compared to and

also explaining the spatial extension of the value of . But, for

this latter point, such a situation implies that the internal gravitic

fields for clusters close together mustnt have a randomly

distributed orientation. In this case, it would mean that for very

large astrophysical structures, there should be coherent

orientations. This constraint could be in agreement with some

published papers (HUTSEMEKERS, 1998; HUTSEMEKERS et al.,

2005) that reveal coherent orientations for large astrophysical

structures (constraint that will have very important consequences

on our solution). This situation is similar, in electromagnetism, to

the situation of magnetic materials, from which the atomic spins

generate a magnetic field at the upper scale of the material

(without an electric field). And I recall that linearized general

relativity leads to the same field equations than Maxwell

idealization. In other words, this situation is consistent with the

general relativity.

So a fourth conclusion is that this case B bis could allow

completely justifying our assumption (I) that explains dark matter

in the frame of general relativity: negligible and maintain of

on large distance.

Conclusion:

We have seen that for the four following constraints, to obtain the

expected order of magnitude of , to obtain the expected

interval of , to obtain a neglected and to be compliant with

some observations, the cluster with its neighbors is the best

candidate, and very likely the only candidate.

We will see below that this origin leads to several very important

predictions (on the orientations of the galaxies spin vector and

the clusters spin vector) that allow testing our solution and

distinguishing our solution from dark matter assumption and from

MOND theories. They are very important because if these

predictions are not verified, our solution will certainly not be the

solution adopted by the nature to explain dark matter. Inversely,

these observations will be difficult to explain in the frame of the

others solutions. In fact, they are not predicted by them.

One can make a general remark. With these calculations, we see

that general relativity requires that some large astrophysical

structures generate significant gravitic fields ( is

significant because sufficient to explain dark matter). And with our

roughly calculation, the own gravitic field of a cluster cannot be

neglected and even it seems to give the right order of magnitude

to explain dark matter. Furthermore, by definition, because

depends on the matter speed of the source

(

) and because the gravitic force also

depends on the speed of matter that undergoes the gravitic field

( ), the influence of gravitic field (compare to gravity

field) must become more and more important with matter speed

(

for ). In general, higher is the scale, higher is the

typical speed. In other words, higher is the scale, greater is the

influence of the gravitic field compared to the gravity field. It is

therefore likely that our assumption (I) is less an assumption than

a necessary condition that must be taken into account at the scale

of galaxies and beyond. We have seen before that at lower scales

than galaxies, the gravitic force was negligible but at the scale of

galaxies, it begins to be significant,

. For a cluster, with our

previous approximation ( ), it would give

.

A last remark on the case D, which is beyond the scope of our

paper, the gravitic field at the scale of the universe could also lead

to an explanation of dark energy but it implies a new fundamental

Fig. 8: Influence of the neighbors on the gravitic field (increase of the

norm of ) and on the gravity field (neutralization of ).

cluster

16

physical assumption (LE CORRE, 2015). Some experiments at CERN

are testing the possibility of such a fundamental assumption.

4.2. Experimental relevancies

If we consider the own gravitic field of the cluster, the maximal

value of this field must be around the center of the masss

distribution of the cluster. So, with our previous deduction on the

origin of the external gravitic field, one can deduce that should

be maximal at the center of the cluster. In other words, our

solution makes a first prediction:

The external gravitic field (our dark matter) should decrease

with the distance to the center of the cluster.

Recent experimental observations (JAUZAC et al., 2014) in

MACSJ0416.1-2403 cluster reveals that the quantity of dark

matter decreases with the distance to the center of the cluster, in

agreement with our origins of external gravitic field. This is the

second main result of our study.

And for the same reason, a second prediction is:

More, we are close to the center of the cluster, more the galaxies

should have dark matter in very unusual and great proportions.

Very recent publications have just revealed the existence of ultra

diffuse galaxies (UDG) in the coma cluster (KODA et al., 2015; VAN

DOKKUM et al., 2015). These unsuspected galaxies show a

distribution concentrated around the clusters center and seem to

have a more important quantity of dark matter than usually.

These UDG would be in agreement with the expected constraint

of our solution.

Our solution leads to another consequence due to the origin of

our . Because our explanation doesnt modify general relativity,

the solution of a declining rotation curve (at the ends of a galaxy)

is always possible. For such a situation, the galaxy must not be

under the influence of the external gravitic field. This means that

the galaxy must be isolated. More precisely, our third prediction

is:

The galaxies without dark matter (with a declining rotation curve)

must be at the ends of the cluster (far from its center) and far

from others clusters (i.e. priori at the borders of a superclusters).

Some observations corroborate this prediction. For example NGC

7793 (CARIGNAN & PUCHE, 1990) has a truly declining rotation

curve. As written in (CARIGNAN & PUCHE, 1990), NGC 7793 is

effectively the most distance member of the Sculptor group (if

the visible member is distributed on the whole group, it also

means that it is at the end of the group) in agreement with our

solution.

I recall that, as said before, if a larger structure than supercluster

(as Laniakea) was the origin of the external gravitic field, the

spatial extension of such a structure would make nearly

impossible to detect galaxies not under the influence of the

expected . So, finding few cases of galaxies with a truly declining

rotation curve is also in agreement with the fact that the good

scale that contributes to the external gravitic field is between the

clusters and the supercluster.

5. Gravitic field of cluster on galaxies

5.1. Application on satellite dwarf galaxies

We are now going to look at satellite dwarf galaxies and retrieve

two unexplained observed behaviors (rotation speed values and

movement in a plane). First, one deduces an asymptotic

expression for the component of gravitic field (our dark matter).

We know that, to explain rotation speed curve, we need two

essential ingredients (internal gravitic field

and embedding

external gravitic field ). Lets calculate how far one has

for our previous studied galaxies.

NGC 5055 NGC 4258 NGC 5033 NGC 2841 NGC 3198 NGC 7331 NGC 2903 NGC 3031 NGC 2403 NGC 247 NGC 4236 NGC 4736 NGC 300 NGC 2259 NGC 3109 NGC 224

The previous values mean that, in our approximation, for

the galactic dynamic is dominated by the external

embedding gravitic field .

And, very far from the center of galaxies, formula can be

written . So far, one can only consider external gravitic

field , formula gives:

Very far from center of galaxies, the speed component

due to internal gravity tends to zero. If we neglect , one

has

In our solution, very far from galaxies center ( ), one

has (if we neglect ):

One can then deduce the rotational speed of satellite galaxies for

which the distances are greater than . In this area, the

own gravitation of galaxies is too weak to explain the measured

rotational speed, but the external gravitic field embedding the

galaxy can explain them. Our previous study gives:

With the relation , one has (in kilo parsec):

It then gives for :

Tab. 1: Distance where internal gravitic field

becomes equivalent

to external gravitic field .

17

This range of rotation speed is in agreement with experimental

measures (ZARITSKY et al., 1997) (on satellite dwarf galaxies)

showing that the high values of the rotation speed continues far

away from the center. This is the third main result of our study.

5.2. About the direction of for a galaxy

Our previous study reveals that, inside the galaxy, external gravitic

field is very small compare to the internal gravitic field

. But

this internal component decreases sufficiently to be neglected far

from galaxy center, as one can see in the following simplified

representation (Fig. 9).

One can then define three areas with specific behavior depending

on the relative magnitude between the two gravitic fields; very far

from center of galaxy for which only direction of acts, close

around center of galaxy for which only direction of acts and a

transition zone for which the two directions act. Just like in

electromagnetism with magnetic field, at the equilibrium, a

gravitic field implies a movement of rotation in a plane

perpendicular to its direction. With these facts, one can make

several predictions.

Very far from center of galaxy, there are the satellite dwarf

galaxies. One can then make a fourth prediction:

The movement of satellite dwarf galaxies, leaded by only

external , should be in a plane (perpendicular to ).

This prediction has been recently verified (IBATA et al., 2014) and

is unexplained at this day. In our solution, it is a necessary

consequence. This is the fourth main result of our study.

One can also make statistical predictions. The external gravitic

field for close galaxies must be relatively similar (in magnitude and

in direction). So, close galaxies should have a relatively similar

spatial orientation of rotations planes of their satellite dwarf

galaxies. The fifth prediction is:

Statistically, smaller is the distance between two galaxies; smaller

is the difference of orientation of their satellite dwarf galaxies

planes.

We saw previously that the cluster of galaxies is likely the good

candidate to generate our external gravitic field . If this

assumption is true, one can expect that the orientations of the

satellite dwarf galaxies planes should be correlated with the

direction of the internal gravitic field of the cluster (which is

our ). It is quite natural (just like for the galaxies) to assume

that, at the equilibrium, the direction of is perpendicular to the

supergalactic plane. It leads to our sixth prediction:

Statistically, inside a cluster, the satellite dwarf galaxies planes

should be close to the supergalactic plane.

These predictions are important because it can differentiate our

solution from the dark matter assumption. Because the dark

matter assumption is an isotropic solution, such an anisotropic

behavior of dwarf galaxies (correlation between dwarf galaxies

directions of two close galaxies and correlation with supergalactic

planes) is not priori expected. As said before, these coherent

orientations could be another clue of a more general property on

very large structures at upper scales. One can recall that there are

some evidences of coherent orientations for some large

astrophysical structures (HUTSEMEKERS, 1998; HUTSEMEKERS et

al., 2005). Furthermore, in our solution, there is no alternative.

Such correlations are imposed by our dark matter explanation.

Very recent observations verify these three predictions (TULLY et

al., 2015). The conclusion is:The present discussion is limited to

providing evidence that almost all the galaxies in the Cen A Group

lie in two almost parallel thin planes embedded and close to

coincident in orientation with planes on larger scales. The two-

tiered alignment is unlikely to have arisen by chance. It means

that the satellites in the Centaurus A group are distributed in

planes (and then also their movement). It is our fourth prediction.

It means that the two planes have very close orientations. It is our

fifth prediction. And it means that these planes have very close

orientations to supergalactic plane. It is our sixth prediction. And

this observation validates that the cluster is likely the good

candidate to generate our external . These observations are

certainly one of the main evidences of the relevance of our

solution (with also the ultra diffuse galaxies). And one can go

further to compare our solution with the non baryonic dark

matter assumption. As we have seen previously, is too small to

influence the orientation of the rotation plane of a galaxy. It

means that our solution implies a decorrelation between the

galaxys rotation planes and the satellite dwarf galaxies planes.

While the galaxys rotation planes are distributed randomly, the

satellite dwarf galaxies planes have a specific orientation imposed

by the clusters orientation. For example in the Centaurus A group,

contrary to their satellite dwarf galaxies planes, the rotation

planes of the galaxies are priori distributed randomly (we will

see that our solution implies that they should be randomly

distributed in a half-space). This situation imposed by our solution

and which is verified by the observation is more difficult to justify

in the traditional dark matter assumption. Because, priori, if non

baryonic matter is distributed as ordinary matter, the gravitation

laws should lead to the same solution, i.e. the same orientation of

the both planes (plane of the galaxy and plane of the satellite

dwarf galaxies). It is not the case, meaning that with this

assumption dark matter and ordinary matter should have

different behaviors. Inversely, if we assume a possible specific

distribution of dark matter to explain the plane of the satellite

dwarf galaxies, it leads to a problem. One must define a specific

distribution for each galaxy (several orientations between the two

planes are possible). For a galaxy, then what is the constraint that

Fig. 9: Evolution of internal gravitic field

compare with external

gravitic field defining three different areas.

18

would favor one configuration over another? Once again a way to

solve this problem would be that baryonic and non baryonic

matters undergo differently the gravitation laws (to converge to

two different mathematical solutions, two different physical

distributions). I recall that one of the main interests of the dark

matter assumption is to not modify the gravitation laws. With

these observations, specific physical laws should make this

assumption a little more complex. Of course, non trivial complex

solutions can certainly be imagined to explain these situations, but

our solution is very natural to explain them. To sum up, these

observations are unexpected in the dark matters assumption but

required by our solution.

One can imagine two others configurations. When the internal

gravitic field of the galaxy and the external is in the opposite

direction:

With this situation (Fig. 10) there are two zones inside the galaxy

with two opposite directions of gravitic field. It leads to the

seventh prediction:

A galaxy can have two portions of its disk that rotate in opposite

directions to each other.

This prediction is in agreement with observations (SOFUE &

RUBIN, 2001).

The other configuration leads to a new possible explanation of the

shape of some galaxies. The warped galaxies could be due to the

difference of direction between internal and external gravitic field,

as one can see in this other simplified 2D representation (Fig. 11).

It would lead to our eighth prediction:

If the gravitic fields would be effectively the main cause of the

wrapping, there should be a correlation between the value of the

wrapping and the angular differences of galaxies and clusters

spins vector.

5.3. About the galaxies spins direction in a cluster

We have seen that in our solution (to maintain on a large area)

the neighbors of a cluster must intervene by adding their own

gravitic field. This vectorial sum can maintain only if the gravitic

fields of the neighbors of a cluster are nearly parallel. This

constraint on our solution leads to a new very important

prediction. If it is not verified, it will be very hard for our solution

to explain dark matter. Our ninth prediction:

The gravitic field of two close clusters must be close to parallel. In

other words, the spin vector of these two close clusters must be

close to parallel.

Once again, this prediction is not trivial (not expected by any other

theories). But some observations seem to make this prediction

possible. For example the results of (HUTSEMEKERS, 1998;

HUTSEMEKERS et al., 2005) reveal coherent orientations for large

astrophysical structures. But furthermore, in the paper of (HU et

al., 2005), their result is that the galaxies spins of the local super

cluster (LSC) point to the center of the LSC. With an hundred of

clusters inside the LSC, one can estimate at least about twenty

clusters along the border of the LSC. It leads to an angle between

two close clusters of around . If the spin of the

cluster follows the same physical influence than the galaxies spins

(point to the center of the LSC), it would mean that the gravitic

fields of two close clusters are nearly parallel, justifying our

previous calculation. Unfortunately, this paper doesnt give results

on the clusters spins.

If the cluster is effectively the origin of and as we have seen it,

if and

are not in the same half-space (inside a cluster), the

galaxy must have two portions of its disk that rotate in opposite

directions to each other. Statistically, these kinds of galaxies are

rare. It leads to the tenth prediction (if it is not checked, our

solution can be considered as inconsistent):

In our solution (from cluster) and

(from galaxy) must be in

the same half-space, for nearly all the galaxies inside a cluster. Say

differently, galaxies spin vector and spin vector of the cluster

(that contains these galaxies) must be in a same half-space.

There are some observations that seem to validate this sine qua

none constraint (HU et al., 2005). Their result, as said before, is

that the galaxies spins of t